This project helps investigators to cope with complex equations, i.e., model biological systems, including the following efforts: 1) Binding rates of hemoglobin (Hb) to various ligands (with M. Perrella). An algorithm has been developed for computing the time course of Hb-ligand biding which allows the resolution of al binding rates. 2) Conformational changes in Hb binding (with K. Vadegriff). Hb-oxygen binding is studied using singular value decomposition (SVD) in an effort to detect conformational changes. 3) Relaxation kinetics of Bacteriorhodopsin (bR) (with R. Hendler, Z. Daneshazy and S. Bose). Our work has shown that both cooperative and noncooperative models are capable of mimicking the features of the relaxation of bR after laser flash. 4) Regression analysis of oxygenation isotherms (with M. Doyle, D. Myers and G. Ackers). Extensive simulations have been run using different assumptions about experimental error, and methods have been devised for generating first estimates of parameters and for solving total least squares equations efficiently. 5) Concentrations of ADP and ATP by partial least squares (PLS) methods (with R. Berger). Several computer codes have been written to implement the use of PLS in the project. 6) Rapid computation of the probability density function (pdf) for the three-phase invariant used in direct methods of phase determination in x-ray crystallography (with U. Shmueli and G. Weiss). It is possible to develop simpler but approximate expressions from the more accurate but time-consuming calculations. This program is presently being implemented and should be completed shortly. 7) Imaging regional cerebral blood flow (with R. Carson). A proposed method for computing this quantity without an explicit measurement of the associated arterial blood flow has been improved upon at PSL. These data are presently being assembled in the Nuclear Medicine Branch to test the utility of the proposed method. 8) Kinetics of reduction of cytochrome aa3 (with R. Hendler and S. Bose). The dynamics of cytochrome reduction are being observed with a rapid-scan multiwavelength spectrophotometer, the output of which is analyzed in terms of methods based on singular-value- decomposition as well as other computational methods.